D. Roberts PHYS 121 University of Maryland Physic² 121: Phundament°ls of Phy²ics I November 20, 2006

D. Roberts PHYS 121 University of Maryland Make-Up Exam 8:00-9:00 –Room PHYS :00-1:00 –Room PHYS :00-6:00 –Room PHYS is a fairly large room (seats ~70)

D. Roberts PHYS 121 University of Maryland Chapter 8 Rotational Equilibrium and Rotational Dynamics

D. Roberts PHYS 121 University of Maryland Torque and Angular Acceleration When a rigid object is subject to a net torque (≠0), it undergoes an angular acceleration The angular acceleration is directly proportional to the net torque –The relationship is analogous to ∑F = ma Newton’s Second Law

D. Roberts PHYS 121 University of Maryland Moment of Inertia The angular acceleration is inversely proportional to the analogy of the mass in a rotating system This mass analog is called the moment of inertia, I, of the object –SI units are kg m 2

D. Roberts PHYS 121 University of Maryland Newton’s Second Law for a Rotating Object The angular acceleration is directly proportional to the net torque The angular acceleration is inversely proportional to the moment of inertia of the object

D. Roberts PHYS 121 University of Maryland More About Moment of Inertia There is a major difference between moment of inertia and mass: the moment of inertia depends on the quantity of matter and its distribution in the rigid object. The moment of inertia also depends upon the location of the axis of rotation

D. Roberts PHYS 121 University of Maryland Moment of Inertia of a Uniform Ring Image the hoop is divided into a number of small segments, m 1 … These segments are equidistant from the axis

D. Roberts PHYS 121 University of Maryland Other Moments of Inertia

D. Roberts PHYS 121 University of Maryland Example, Newton’s Second Law for Rotation Draw free body diagrams of each object Only the cylinder is rotating, so apply  = I  The bucket is falling, but not rotating, so apply  F = m a Remember that a =  r and solve the resulting equations Problem: A solid, frictionless cylindrical reel of mass M = 3 kg and radius R = 0.4 m is used to draw water from a well. A bucket of mass 2 kg is attached to a cord wrapped around the cylinder. –Find the tension in the cord and acceleration of the bucket

? Two cylinders of the same size and mass roll down an incline. Cylinder A has most of its weight concentrated at the rim, while cylinder B has most its weight concentrated at the center. Which reaches the bottom of the incline first? A 2.B 3.Both reach at the same time 0 of 5 30

D. Roberts PHYS 121 University of Maryland Rotational Kinetic Energy An object rotating about some axis with an angular speed, ω, has rotational kinetic energy ½Iω 2 Energy concepts can be useful for simplifying the analysis of rotational motion

D. Roberts PHYS 121 University of Maryland Total Energy of a System Conservation of Mechanical Energy –Remember, this is for conservative forces, no dissipative forces such as friction can be present –Potential energies of any other conservative forces could be added

D. Roberts PHYS 121 University of Maryland Work-Energy in a Rotating System In the case where there are dissipative forces such as friction, use the generalized Work- Energy Theorem instead of Conservation of Energy W nc =  KE t +  KE R +  PE